℘⁡(z) and ζ⁡(z) are meromorphic
functions with poles at the lattice points. ℘⁡(z) is even
and ζ⁡(z) is odd. The poles of
℘⁡(z) are double with residue 0; the poles of
ζ⁡(z) are simple with residue 1. The function
σ⁡(z) is entire and odd, with simple zeros at the
lattice points. When it is important to display the lattice with the functions
they are denoted by ℘⁡(z|𝕃), ζ⁡(z|𝕃),
and σ⁡(z|𝕃), respectively.